Question

If A is a square matrix of order 3 and $|A| = 6$, then the value of $|\text{adj} A|$ is:

• A. 6
• B. 36
• C. 27
• D. 216

Hint:

For any square matrix, $|\text{adj} A| = |A|^{n-1}$. So, just find the order $n$ and plug in!

Answer 1

The Correct Option is B

For any square matrix $A$ of order $n$, the determinant of its adjugate (or adjoint) is given by: \[ |\text{adj} A| = |A|^{n-1}. \] This property comes from the fact that the adjugate matrix is formed by the cofactors, and its determinant is the original determinant raised to the power $(n-1)$. Given: \[ |A| = 6, n = 3. \] So, \[ |\text{adj} A| = |A|^{3-1} = |A|^2 = 6^2 = 36. \] Therefore, the value of $|\text{adj} A|$ is 36.